Note di Matematica 26, n. 2, 2006, 147–152.
A Darboux theorem for generalized contact
manifolds i
David E. Blair
Department of Mathematics, Michigan State University,
East Lansing, MI 48824, USA
[email protected]
Luigia Di Terlizzi
Dipartimento di Matematica, Università di Bari
Via Orabona 4, 70125 Bari, Italy
[email protected]
Jerzy J. Konderakii
Dipartimento di Matematica, Università di Bari
Via Orabona 4, 70125 Bari, Italy
Received: 26/10/2005; accepted: 28/10/2005.
Abstract. We consider a manifold M equipped with 1-forms η1 , . . . , ηs which satisfy certain
contact like properties. We prove a generalization of the classical Darboux theorem for such
manifolds.
Keywords: contact manifold, Darboux theorem, parallelized null bundle
MSC 2000 classification: 53D22, 53D10
1
Introduction
In the present paper we consider a smooth manifold M of dimension (2n+s)
admitting a set of 1-forms η1 , . . . , ηs such that: η1 ∧ · · · ∧ ηs ∧ (dη1 )n is nowhere
vanishing, the form dη1 is of constant rank equal to 2n and dη1 = · · · = dηs .
We call such manifolds generalized contact manifolds with a parallelizable null
bundle. We observe that such manifolds carry a transverse symplectic foliation
F determined by the null distribution of dη1 . The idea of our proof is to apply
the Darboux theorem for the transverse part of the foliation and then extend
it along the leaves. Our version of the Darboux theorem is clearly local and
may be expressed as a property of certain forms on an open subset of R2n+s .
However, we present here this theorem in the framework of the generalized
i
Research supported by the Italian MIUR 60% and GNSAGA
The first two authors express their sadness due to the death of Professor Konderak on 14
September 2005.
ii
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D. E. Blair, L. Di Terlizzi, J. J. Konderak
contact manifolds with a parallelized null bundle and almost S-manifolds to
underline its geometric applications.
The notion of the generalized contact manifold with a parallelized null bundle arises in the context of the so called (almost) S-manifolds. These are manifolds which, besides the 1-forms η1 , . . . , ηs , are equipped with an f -structure, a
compatible Riemannian metric and appropriate integrability conditions on the
f -structure, cf. [2]. The (almost) S-manifolds were studied by various authors
and are currently under intensive research, cf. [4] and the references there.
The classical theorems of Darboux for symplectic and contact manifolds are
the starting point for study of these manifolds. Hence there are many possible
references. The standard proofs and some generalizations may be found for
example in [5, 6, 8, 11]; the modern approach including the so called Moser’s
trick may be found in [7]. Further generalizations are given in [1, 10, 12].
All manifolds, maps, distributions considered here are smooth i.e. of the
class C ∞ ; we denote by Γ(−) the set of all sections of a corresponding bundle.
We use the convention that 2u ∧ v = u ⊗ v − v ⊗ u.
2
Main Theorem
Throughout all of this paper we assume that there is given a (2n + s)dimensional manifold M with 1-forms η1 , . . . , ηs such that dη1 = · · · = dηs =: F
and
η1 ∧ · · · ∧ ηs ∧ F n 6= 0
(1)
at each point of M . We suppose also that rank(F ) = const. = 2n. Then we
call (M, ηi ), (i = 1, . . . , s) a generalized
contact manifold with a parallelizable
T
null bundle. We denote by D := si=1 ker ηi . Then from (1) it follows that D is a
vector subbundle of T M of rank 2n and F restricted to D is non-degenerate, i.e.
(D, F |D×D ) is a symplectic vector bundle over M . By Null(Fp ) we denote the
set of all X ∈ Tp M , (p ∈ M ), such that XyF = 0; it is called the null subspace
associated
with Fp . Then the null subbundle associated with F is Null(F ) :=
S
Null(Fp ) where the union is taken over all p ∈ M . Null(F ) is a vector bundle
over M of rank s. Moreover, we have that Null(F ) ⊕ D = T M .
In the following lemma we generalize the existence theorem of the Reeb
vector field on contact manifolds.
1 Lemma. There exist unique ξ1 , . . . , ξs ∈ Γ(Null(F )) such that ηi (ξj ) = δij
for all i, j = 1, . . . , s. Moreover, [ξi , ξj ] = 0, Lξi X ∈ Γ(D) and Lξi ηj = 0 for all
i, j = 1, . . . , s, and X ∈ Γ(D).
Proof. We consider a section Ψ of the bundle (Null(F ))∗ ⊗ Rs given by
Ψp (X) := (η1 (X), . . . , ηs (X)) for each p ∈ M and each X ∈ Null(Fp ). Since (1)
A Darboux theorem for generalized contact manifolds
149
holds and dim(Null(Fp )) = s, it follows that Ψp is an isomorphism between
Null(Fp ) and Rs . Then we put
(ξi )p := Ψ−1
p (ei )
(2)
where e1 , . . . , es is the canonical basis of Rs and p varies over all of M . Such
fields ξi are smooth and satisfy ηi (ξj ) = δij for all i, j ∈ { 1, . . . , s }. Then the
uniqueness of the existence of ξ1 , . . . , ξs follows from the fact they are uniquely
defined by condition (2).
We have that for each k ∈ { 1, . . . , s }
ηk ([ξi , ξj ]) = −2F (ξi , ξj ) + ξi ηk (ξj ) − ξj ηk (ξi ) = 0
[ξi , ξj ]yF = Lξi (ξj yF ) − ξj y(Lξi F ) = −ξj y(ξi ydF + d(ξi yF )) = 0.
This implies that [ξi , ξj ] annihilates ηk and F for all k. Therefore [ξi , ξj ] vanishes.
If X ∈ Γ(D) and i, j = 1, . . . , s, then [ξi , X]yηj = −2F (ξi , X) + ξi ηj (X) −
Xηj (ξi ) = 0 hence it follows that [ξi , X] ∈ Γ(D). Since
Lξi ηj (ξk ) = ξi (ηj (ξk )) − ηj ([ξi , ξk ]) = 0
(Lξi ηj )(X) = ξi (ηj (X)) − ηj ([ξi , X]) = 0
QED
for all i, j, k ∈ { 1, . . . , s } and X ∈ Γ(D), it follows that Lξi ηj = 0.
We shall call ξ1 , . . . , ξs the Reeb vector fields associated with the structure
(M, ηi ), (i = 1, . . . , s).
The generalized contact structure with a parallelized null bundle (M, ηi ),
(i = 1, . . . , s) may be characterized in another way. If Cs (M ) := M × Rs+ is the
s-cone over M where R+ denotes the positive real numbers and r = (r1 , . . . , rs )
then we put
!
s
s
X
X
2
2
2ri dri ∧ ηi .
ri ηi = krk F +
ωF := d
i=1
i=1
It is straightforward to observe that (Cs (M ), ωF ) is a symplectic manifold, cf. [3].
2 Remark. There exists on D a Riemannian metric g0 and a compatible
complex structure J0 such that dη1 is the associated Kähler form; actually J0
may be obtained via the polar decomposition of dη1 with respect to any Riemannian metric. Then we extend g0 to a Riemannian metric g on M by assuming
that g|D = g0 , ξ1 , . . . , ξs are orthonormal and D ⊥ Null(F ). We extend also J0
to an endomorphism φ of T M by assuming that φ|D = J0 and φ|Null(F ) = 0. As
a result we obtain the set (M, g, φ, ηi , ξj ), (i, j = 1, . . . , s) which is an almost Smanifold. If a certain integrability condition, the so called normality condition,
for φ is satisfied, then the manifold is called an S-manifold, cf. [2, 4].
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D. E. Blair, L. Di Terlizzi, J. J. Konderak
3 Theorem (Darboux Theorem). Let (M 2n+s , ηi ), (i . . . , s), be a generalized contact manifold with a parallelized null bundle, then locally on M there
exist coordinates (x1 , . . . , xn , y1 , . . . , yn , z1 , . . . , zs ) such that for each i = 1, . . . , s
ηi = dzi +
n
X
xα dy α .
α=1
Proof. From the Lemma we get the Reeb vector fields ξ1 , . . . , ξs . We denote
by F the foliation on M determined by the distribution span{ ξ1 , . . . , ξs } =
Null(F ). Since ξi yF = 0 and dF = 0 then it follows that F is a basic 2-form
on M , cf. [9]. Since D is complementary to T F and (D, F |D×D ) is a symplectic
vector bundle then the foliated manifold (M, F, F ) is transverse symplectic,
cf. [9, p.53]. Let p0 ∈ M then there exists an open connected neighbourhood
U of p0 and a trivial fibration πU : U → V on a 2n-dimensional symplectic
manifold (V, ω) with πU∗ (ω) = F and such that the fibres of πU coincide with
the intersection of F with U , cf. [9]. From the classical Darboux theorem, after
possible compression of V , we may assume that V is an open neighbourhood of
0 ∈ R2n with πU (p0 ) = 0 and
!
n
n
X
X
x̃α dỹα
dx̃α ∧ dỹα = d
ω=
α=1
α=1
where (x̃1 , . . . , x̃n , ỹ1 , . . . , ỹn ) are the canonical coordinates on V ⊂ R2n . We
denote by (x1 , . . . , xn , y1 , . . . , yn ) := πU∗ (x˜1 , . . . , x̃n , ỹ1 , . . . , ỹn ) which are coordinates on the transverse part to F. Let σ : V → U be a smooth section of πU
such that σ(0) = p0 .
Let ψti be a 1-parameter subgroup of local transformations associated with
ξi for each i ∈ { 1, . . . , s }. Since [ξi , ξj ] = 0 then ψti1 and ψtj2 commute with each
other within their domains. Then we define a map Φ from a neighbourhood of
0 ∈ R2n+s into M by setting Φ(x, y, z̃) := ψz1˜1 ◦ · · · ◦ ψz̃ss (σ(x, y)) where x stands
for (x1 , . . . , xn ), y stands for (y1 , . . . , yn ) and z̃ stands for (z̃1 , . . . , z̃s ). Since Φ
is smooth and Φ(0) = p0 , there exists an open subset W of 0 ∈ R2n+s such that
Φ(W ) ⊂ U . It follows from the construction of Φ that
d0 Φ ∂∂z̃i |0 = (ξi )p0 , d0 Φ ∂x∂α |0 = d0 σ ∂x∂α |0 , d0 Φ ∂y∂α |0 = d0 σ ∂y∂α |0
for each i ∈ { 1, . . . , s } and α ∈ { 1, . . . , n }; here d0 Φ and d0 σ denote the
differential of the corresponding maps taken at 0. Since (ξi )p0 , (i = 1, . . . , s) give
a basis of the vertical space of πU and d0 σ( ∂x∂α |0 ), d0 σ( ∂y∂β |0 ), (α, β = 1, . . . , n)
give a basis of the horizontal space, it follows that d0 Φ is an isomorphism and Φ is
a diffeomorphism in a neighbourhood of 0 ∈ R2n+s . Hence we may assume, after
151
A Darboux theorem for generalized contact manifolds
possible compression of W , that Φ : W → Φ(W ) is a diffeomorphism of W onto
Φ(W ) ⊂ U an open neighbourhood of p0 . From the construction of Φ it follows
that for each i = 1, . . . , s we have that Φ∗ ( ∂∂z̃i ) = ξi . We denote by ϕ the inverse
of Φ i.e. ϕ = Φ−1 : Φ(W ) → W . The map ϕ = (x1 , . . . , xn , y1 , . . . , yn , z̃1 , . . . , z˜s )
is a chart on M with respect to which ξi = ∂∂z̃i and dz˜i (ξj ) = δij for all i, j ∈
{ 1, . . . , s }. The 1-form
n
X
xα dyα
(3)
ηi − dz̃i −
α=1
is defined on Φ(W ) and is closed for each i = 1, . . . , s since dηi = F . From the
Poincaré lemma, after possible compression on W , we may assume that there
exist smooth functions f1 , . . . , fs ∈ C ∞ (Φ(W )) such that
dfi = ηi − dz̃i −
n
X
xα dyα
(4)
α=1
for all i ∈ { 1, . . . , s }. Since the 1-forms in (3) are basic with respect to the
foliation F so are dfi for all i. This means that ∂∂fz̃ji = 0 for all i, j = 1, . . . , s.
Then we consider the new coordinates
xα := xα , yβ := yβ , zi := z̃i + fi
(5)
where α, β ∈ { 1, . . . , n } and i ∈ { 1, . . . , s }. We need to comment that equations
(5) actually define new coordinates; in fact the Jacobian matrix between ϕ and
the new coordinates is the following


(δαβ )
∂fi
)
−( ∂x
α

.
(δαβ )
∂fi
−( ∂yα ) (δij )
(6)
Therefore our assertion follows from the definition of the coordinates (x, y, z) in
QED
(5) and equation (4).
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